Suppose \(x\) is a solution to \(x^2 + 1 = 7x\) . What is the sum of \(x\) and its reciprocal?
+1252 You can simply find x and calculate
Eventually, this equals 7.
However, there is a faster way. Can you find out yourself?
You are very welcome!
:P
I think using the quadratic formula, we can find what x is equal to.
so
x^2+1=7x
Subtract 7x from both sides
x^2-7x+1
(Notice it is in the form ax^2+bx+c)
Apply the quadratic formula
\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
You will eventually get
x=6.85410196 approx.=6.85
Now I don't understand what do you mean what is the sum of x?
Its reciprocal is just 1/6.85 i guess.
+129852 Rearrange as
x^2 - 7x = -1
Complete the square on x
x^2 - 7x + 49/4 = -1 + 49/4
(x - 7/2)^2 = 45/4 take the positive root
x - 7/2 = √45 / 2
x - 7/2 = 3√5/2
x = [ 7 + 3√5] / 2
Its reciprocal is 2 / [ 7 + 3√5]
So
[ 7 + 3√5] / 2 + 2 / [ 7 + 3√5] =
[ 7 + 3√5] ^2 + 4
_______________ =
2 [ 7 + 3√5]
[ 49 + 42√5 + 45 + 4 ]
__________________ =
2 [ 7 + 3√5]
[ 98 + 42 √5 ]
____________ =
2 [ 7 + 3√5]
49 + 21√5
_________ =
7 + 3√5
[49 + 21√5] [ 7 - 3√5]
__________________ =
49 - 45
343 + 147√5 - 147√5 - 63*5
_______________________ =
4
343 - 315
_________ =
4
28
__ =
4
7
